We saw in §8.4 that an LTI filter is stable if and only
if all of its poles are strictly inside the unit circle (
) in
the complex
plane. In particular, a pole
outside the unit
circle (
) gives rise to an impulse-response component
proportional to
which grows exponentially over time
. We
also saw in §6.2 that the z transform of a growing exponential does
not converge on the unit circle in the
plane. However, this
was the case for a causalexponential
, where
is the unit-step function (which switches from 0 to 1 at time 0). If
the same exponential is instead anticausal, i.e., of the form
, then, as we'll see in this section, its z transform does exist on
the unit circle, and the pole is in exactly the same place as in the
causal case. Therefore,to unambiguously invert a z transform, we must know
its region of convergence. The critical question is whether
the region of convergence includes the unit circle: If it does, then
each pole outside the unit circle corresponds to an anticausal, finite
energy, exponential, while each pole inside corresponds to the usual
causal decaying exponential.